Collision Time in Adiabatic Expansion – Rankers Physics
Topic: Thermal Physics
Subtopic: Thermodynamics

Collision Time in Adiabatic Expansion

Consider an ideal gas confined in an isolated closed chamber. As the gas undergoes an adiabatic expansion, the average time of collision between molecules increases as \(V^q\), where V is the volume of the gas. The value of q is \(\left(\gamma = \frac{C_p}{C_v}\right)\)
\(\frac{3\gamma + 5}{6}\)
\(\frac{3\gamma - 5}{6}\)
\(\frac{\gamma + 1}{2}\)
\(\frac{\gamma - 1}{2}\)

Solution:

Mean free path \(\lambda \propto V\) and \(v_{\text{rms}} \propto \sqrt{T}\). For an adiabatic process, \(T \propto V^{-(\gamma - 1)}\), so \(v_{\text{rms}} \propto V^{-(\gamma - 1)/2}\). Average time \(\tau = \frac{\lambda}{v_{\text{rms}}} \propto V^{1 + (\gamma - 1)/2} = V^{(\gamma+1)/2}\), so \(q = \frac{\gamma+1}{2}\).

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